Subgroups — #14573174

Order 2432.

504 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597396	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927113	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926766	C₂	1	2	yes	maximal, complemented
5	#11926759	C₂	1	2	yes	maximal, complemented
6	#11926744	C₂	1	2	yes	maximal, complemented
7	#11926739	C₂	1	2	yes	maximal, complemented
8	#11926571	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926543	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10597266	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×#64541	C₂²	1	1	yes	characteristic
14	C₂²×D₇₆	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
16	C₄×D₄×C₁₉	—	2	2	no	—
17	C₄×D₄×C₁₉	C₂²	1	2	yes	complemented
18	C₁₉×#117	—	2	1	no	—
19	C₁₉×#117	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	#10597396	—	2	4	no	—
23	#10597395	—	2	4	no	—
24	C₂×#64528	C₂²	1	1	yes	characteristic, complemented
25	C₂×#64527	C₂²	1	2	yes	—
26	C₂×#64527	C₂²	1	1	yes	characteristic, complemented
27	C₂×#64526	C₂²	1	1	yes	characteristic, complemented
28	C₂×#64525	C₂²	1	2	yes	—
29	C₂×#64525	C₂²	1	1	yes	characteristic, complemented
30	C₂×#64520	C₂²	1	1	yes	characteristic
31	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
32	#10597341	—	2	1	no	—
33	#10597341	—	2	1	no	—
34	#10597328	—	2	2	no	—
35	#10597321	C₂²	1	2	yes	—
36	#10597320	C₂²	1	2	yes	—
37	#10597267	C₂²	1	2	yes	complemented
38	#10597266	C₂²	1	2	yes	—
39	#10597266	C₂²	1	2	yes	—
40	#10597266	C₂²	1	2	yes	complemented
41	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	complemented
42	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	complemented
43	C₂³×D₁₉	—	2	1	no	—
44	#41×C₁₉	—	2	2	no	—
45	#41×C₁₉	D₄	1	4	yes	complemented
46	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₉	—	2	1	no	—
48	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	1	no	—
50	C₂²×C₄×C₁₉	—	2	1	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₉	—	2	1	no	—
54	#64541	—	2	2	no	—
55	#64541	—	2	2	no	—
56	C₂×D₇₆	—	2	2	no	—
57	C₂×D₇₆	—	2	2	no	—
58	C₂×D₇₆	—	2	1	no	—
59	C₂×D₇₆	—	2	2	no	—
60	C₂×D₇₆	—	2	2	no	—
61	C₂×D₇₆	C₂³	1	2	yes	—
62	C₂×D₇₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₉	—	2	1	no	—
64	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₉	—	2	1	no	—
66	C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₁₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
72	C₄²×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	C₂³	1	2	yes	—
74	#64528	—	2	2	no	—
75	#64527	—	2	4	no	—
76	#64527	—	2	2	no	—
77	#64526	—	2	2	no	—
78	#64525	—	2	4	no	—
79	#64525	—	2	2	no	—
80	(C₂×D₁₉)⋊C₄	—	2	2	no	—
81	(C₂×D₁₉)⋊C₄	—	2	2	no	—
82	#64520	—	2	2	no	—
83	C₂×C₁₉⋊C₈	—	2	2	no	—
84	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
85	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
87	C₂³×C₁₉	—	2	1	no	—
88	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×D₁₉	—	2	2	no	—
90	C₂²×D₁₉	—	2	1	no	—
91	C₂²×D₁₉	—	4	2	no	—
92	C₂²×D₁₉	—	8	1	no	—
93	C₂²×D₁₉	—	8	1	no	—
94	Q₈×C₁₉	—	2	1	no	—
95	Q₈×C₁₉	C₂×D₄	1	2	yes	—
96	D₄×C₁₉	—	2	4	no	—
97	D₄×C₁₉	—	2	1	no	—
98	D₄×C₁₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	8	2	no	—
103	C₂×C₄×C₁₉	—	8	1	no	—
104	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#41	1	2	yes	—
109	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₁₉	—	4	2	no	—
111	C₂×Dic₁₉	—	2	1	no	—
112	D₇₆	—	2	2	no	—
113	D₇₆	—	2	1	no	—
114	D₇₆	—	2	2	no	—
115	D₇₆	—	2	1	no	—
116	D₇₆	—	4	2	no	—
117	D₇₆	—	8	2	no	—
118	D₇₆	—	8	2	no	—
119	C₄×D₁₉	—	4	2	no	—
120	Dic₃₈	—	2	1	no	—
121	Dic₃₈	—	2	1	no	—
122	C₁₉⋊C₈	—	2	1	no	—
123	C₁₉⋊C₈	—	2	1	no	—
124	C₁₉⋊C₈	—	2	2	no	—
125	#3045	—	19	1	no	maximal, Sylow
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	1	no	—
128	C₂²×C₁₉	—	2	2	no	—
129	C₂²×C₁₉	—	4	1	no	—
130	C₂²×C₁₉	—	4	1	no	—
131	C₂²×C₁₉	#121	1	2	yes	—
132	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₁₉	—	4	1	no	—
134	C₂×D₁₉	—	4	2	no	—
135	C₂×D₁₉	—	4	1	no	—
136	C₂×D₁₉	—	4	2	no	—
137	C₂×D₁₉	—	4	1	no	—
138	C₂×D₁₉	—	4	2	no	—
139	C₂×D₁₉	—	8	4	no	—
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	4	2	no	—
144	C₄×C₁₉	#121	1	2	yes	—
145	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
147	Dic₁₉	—	4	1	no	—
148	C₂×AGL(1,ℤ/8)	—	19	1	no	—
149	C₂×#135	—	19	1	no	—
150	#535	—	19	1	no	—
151	#520	D₁₉	1	1	yes	characteristic, complemented
152	#464	—	19	2	no	—
153	#463	—	19	2	no	—
154	#460	—	19	2	no	—
155	#459	—	19	2	no	—
156	#424	—	19	1	no	—
157	#417	—	19	1	no	—
158	C₂×#102	—	19	1	no	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	—	4	2	no	—
161	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
162	C₂×C₁₉	#576	1	1	yes	characteristic
163	C₂×C₁₉	#577	1	1	yes	characteristic
164	D₁₉	—	8	1	no	—
165	D₁₉	—	8	2	no	—
166	C₂×#41	—	19	1	no	—
167	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
168	C₂²×D₄	—	19	1	no	—
169	AGL(1,ℤ/8)	—	38	4	no	—
170	#135	—	38	4	no	—
171	C₂×Q₁₆	—	19	1	no	—
172	C₂×SD₁₆	—	19	2	no	—
173	C₂×SD₁₆	—	19	1	no	—
174	C₂×SD₁₆	—	19	1	no	—
175	C₂×D₈	—	19	2	no	—
176	C₂×D₈	—	19	1	no	—
177	C₂×M₁₆	—	19	1	no	—
178	C₂²×C₈	—	19	1	no	—
179	#127	—	19	2	no	—
180	#124	—	19	2	no	—
181	#121	—	38	1	no	—
182	#121	—	38	1	no	—
183	#120	—	38	2	no	—
184	C₄×Q₈	C₂×D₁₉	1	2	yes	complemented
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	C₂×D₁₉	1	2	yes	complemented
187	#117	—	2	1	no	—
188	#117	C₂×D₁₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
191	C₄⋊C₈	—	19	2	no	—
192	C₄⋊C₈	—	19	2	no	—
193	#103	—	19	2	no	—
194	#102	—	19	2	no	—
195	#102	—	19	2	no	—
196	#102	—	19	2	no	—
197	C₁₉	#3045	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	38	1	no	—
199	#41	—	2	2	no	—
200	#41	—	38	2	no	—
201	#41	—	38	2	no	—
202	#41	D₇₆	1	4	yes	complemented
203	C₂×Q₈	—	19	1	no	—
204	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	19	2	no	—
207	C₂×D₄	—	19	1	no	—
208	C₂×D₄	—	38	1	no	—
209	C₂×D₄	—	38	2	no	—
210	C₂×D₄	—	38	2	no	—
211	C₂×D₄	—	38	1	no	—
212	C₂×D₄	—	38	2	no	—
213	C₂×D₄	—	38	2	no	—
214	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	38	1	no	—
219	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
220	Q₁₆	—	38	2	no	—
221	SD₁₆	—	38	2	no	—
222	SD₁₆	—	38	4	no	—
223	SD₁₆	—	38	2	no	—
224	D₈	—	38	2	no	—
225	D₈	—	38	4	no	—
226	M₁₆	—	38	2	no	—
227	C₂×C₈	—	19	1	no	—
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	19	2	no	—
230	C₂×C₈	—	38	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
234	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
235	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	38	2	no	—
238	C₂²⋊C₄	—	38	2	no	—
239	C₂²⋊C₄	C₂²⋊D₁₉	1	2	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	C₂²×D₁₉	1	2	yes	—
242	C₂³	—	2	1	no	—
243	C₂³	—	38	2	no	—
244	C₂³	—	38	1	no	—
245	C₂³	—	76	2	no	—
246	C₂³	—	152	1	no	—
247	C₂³	—	152	1	no	—
248	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	38	1	no	—
251	Q₈	—	38	1	no	—
252	Q₈	C₂×D₇₆	1	2	yes	—
253	D₄	—	2	4	no	—
254	D₄	—	2	1	no	—
255	D₄	—	38	2	no	—
256	D₄	—	38	1	no	—
257	D₄	—	38	2	no	—
258	D₄	—	38	1	no	—
259	D₄	—	76	2	no	—
260	D₄	—	76	2	no	—
261	D₄	—	152	2	no	—
262	D₄	—	152	2	no	—
263	D₄	C₂×D₇₆	1	2	yes	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	38	1	no	—
270	C₂×C₄	—	76	2	no	—
271	C₂×C₄	C₂×D₇₆	1	2	yes	—
272	C₂×C₄	#64541	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
274	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
277	C₈	—	38	1	no	—
278	C₈	—	38	1	no	—
279	C₈	—	38	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	76	1	no	—
286	C₂²	—	76	2	no	—
287	C₂²	—	76	1	no	—
288	C₂²	—	76	2	no	—
289	C₂²	—	76	1	no	—
290	C₂²	—	76	2	no	—
291	C₂²	—	152	4	no	—
292	C₂²	#10597376	1	2	yes	—
293	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	76	1	no	—
299	C₄	#10597376	1	2	yes	—
300	C₄	C₂²×D₇₆	1	1	yes	characteristic
301	C₄	C₂×#64541	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	152	2	no	—
305	C₂	—	152	1	no	—
306	C₂	C₂×#10597376	1	1	yes	characteristic
307	C₂	#11927347	1	1	yes	characteristic
308	C₂	#11927352	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#520	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#520	D₁₉
hypocenter	C₁₉	#3045
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#10597395	C₂	semidirect
2. C₂×#10597396	C₂	semidirect
3. #11927113	C₂	semidirect
4. #11926766	C₂	semidirect
5. #11926759	C₂	semidirect
6. #11926744	C₂	semidirect
7. #11926739	C₂	semidirect
8. #11926571	C₂	semidirect
9. #11926543	C₂	semidirect
10. C₂×#10597266	C₂	semidirect
11. C₁₉×#520	C₂	semidirect
15. C₄×Q₈×C₁₉	C₂²	semidirect
17. C₄×D₄×C₁₉	C₂²	semidirect
24. C₂×#64528	C₂²	semidirect
26. C₂×#64527	C₂²	semidirect
27. C₂×#64526	C₂²	semidirect
29. C₂×#64525	C₂²	semidirect
37. #10597267	C₂²	semidirect
40. #10597266	C₂²	semidirect
41. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
42. C₄⋊(C₁₉⋊C₈)	C₂²	semidirect
45. #41×C₁₉	D₄	semidirect
151. #520	D₁₉	semidirect
184. C₄×Q₈	C₂×D₁₉	semidirect
186. C₄×D₄	C₂×D₁₉	semidirect
197. C₁₉	#3045	semidirect
202. #41	D₇₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#10597395	2
C₂×#10597396	2
#11927113	2
#11926766	2
#11926759	2
#11926744	2
#11926739	2
#11926571	2
#11926543	2
C₂×#10597266	2
C₁₉×#520	2
#3045	19